Average Error: 58.9 → 3.1
Time: 33.5s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\frac{1}{a} + \frac{1}{b}\]
\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}
↓
\frac{1}{a} + \frac{1}{b}
double f(double a, double b, double eps) {
        double r4497103 = eps;
        double r4497104 = a;
        double r4497105 = b;
        double r4497106 = r4497104 + r4497105;
        double r4497107 = r4497106 * r4497103;
        double r4497108 = exp(r4497107);
        double r4497109 = 1.0;
        double r4497110 = r4497108 - r4497109;
        double r4497111 = r4497103 * r4497110;
        double r4497112 = r4497104 * r4497103;
        double r4497113 = exp(r4497112);
        double r4497114 = r4497113 - r4497109;
        double r4497115 = r4497105 * r4497103;
        double r4497116 = exp(r4497115);
        double r4497117 = r4497116 - r4497109;
        double r4497118 = r4497114 * r4497117;
        double r4497119 = r4497111 / r4497118;
        return r4497119;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r4497120 = 1.0;
        double r4497121 = a;
        double r4497122 = r4497120 / r4497121;
        double r4497123 = b;
        double r4497124 = r4497120 / r4497123;
        double r4497125 = r4497122 + r4497124;
        return r4497125;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.9
Target14.3
Herbie3.1
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 58.9

    \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

  2. Simplified27.9

    \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}{\mathsf{expm1}\left(\varepsilon \cdot b\right)} \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}}\]

  3. Taylor expanded around 0 3.1

    \[\leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}}\]

  4. Final simplification3.1

    \[\leadsto \frac{1}{a} + \frac{1}{b}\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :pre (and (< -1 eps) (< eps 1))

  :herbie-target
  (/ (+ a b) (* a b))

  (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))))